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in the 1940s the word “boner” used to mean “huge mistake” and it still pretty much means that
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HUP, EPR, and Bell’s Theorem
Abstract
An educational document discussing the Heisenberg Uncertainty Principle, the EPR (Einstein- Podolsky-Rosen) Paradox, and Bell’s Theorem, written for an audience without a background in physics, but with their head still screwed on right.
1 Introduction
Ah, quantum mechanics. A bizarre theory which unfortunately describes our physical world exceed- ingly well. Einstein didn’t get it. Bohr didn’t get it. I don’t get it. And soon, you won’t get it either. As the saying goes, the more you know about quantum mechanics, the less you understand it.
I will be skipping around in terms of topics covered in undergraduate quantum mechanics courses to prepare you for the actual beast, Entanglement.
Entanglement, the property of quantum systems to remain correlated even when separated, is a concept which has transformed from a worrisome byproduct of a thought experiment [1] into a cornerstone of quantum mechanics itself. What is a quantum mechanics? Google is your friend, my dear reader. My time with you is limited„ and I cannot teach you the alphabet to make you read Shakespeare. I can only explain what you directly need to understand this article. Anything else shall be your homework, and if I am feeling kind at the end, I will provide a list of accessible resources on learning quantum mechanics the RIGHT way.
As we dive into the frankly confusing world of entanglement, it is vital that you remember one thing– A quantum particle is described by a wave function, Ψ. This wave function is a solution to the Schrodinger equation.
This is what they mean when they say something is both a particle and a wave; It’s behavior can be described by a special kind of wave equation, which we all know and love as the Schrodinger Wave Equation. But that’s not important right now. I’ll explain more if I need to. We need to get to HUP.
2 Heisenberg’s Uncertainty Principle
Formulated by Werner Heisenberg in 1927, the Uncertainty Principle is an indomitable tenet in the field of quantum mechanics. Its premise is simple. The more precisely a particle’s momentum is determined, the less precisely is its momentum. In one dimension, this can be summarized with the following mathematical statement: ∆x∆p ≤ 2
Here, ∆x is the standard deviation or “spread” of the position x, while ∆p is the standard deviation of the momentum p. As the spread of one quantity decreases, the other must increase in order to maintain the inequality. I will not bother proving the Principle in full, but I have Heisenberg’s original proof in the references.
Is that it?
Ummmmm, no. An important thing to remember about HUP is that it is not exclusive to x and p. HUP applies to any two quantum mechanical operators, A, B, which do not commute with each other i.e. [A, B] = AB − BA = 0. But that’s all mathematical nonsense, Min! What does it really mean?
Fine! I’m only doing this because it will be useful when we get to measurements in the EPR paradox and Bell theorems. In order to understand what “not commuting” means in the physical sense, let’s use our favorites, position and momentum, as an example. In quantum mechanics, xˆ andpˆ are referred to as the position and momentum operators respectively. (Why the little hats? Firstly, they’re cute, and secondly, well, you’ll see.) The whole point of calling them operators is that they act on wave functions. And in the crudest sense possible (please don’t try this at home, folks), hitting an operator on a wave function and taking the expectation value, gives a measurement of the quantum mechanical system.
There is about three semesters of quantum mechanical education I’m waving off right now, but bear with me. When we act the momentum operator on the system, in some sense we extract the momentum. Same thing for position. However, the whole deal about x and p is that they do not commute. So, the order in which you conduct the measurements absolutely does matter. First measuring x and then p would give you a different answer than first measuring p and then x. This is because the very act of measuring a quantum state changes it. That’s right! It changes. This makes all the difference when you consider the standard deviation of a bunch of measurements. If my memory of introductory quantum mechanics serves me right, after about three pages of algebra you arrive at the familiar position-momentum uncertainty principle.
The moral of the story is that the non-commutativity of these operators manifests as a sort of granularity in the accuracy of measurements you can make on a physical system. This granularity is retained between any other kinds of non-commuting measurements you can make!
On second thought, do you really need this? Probably not. But, the algebra of uncertainty principles is a pet project to me. Especially the strangest of them all, the energy-time uncertainty principle. Enough on that! Here’s the main takeaway (other than the actual HUP statement) that you need from this section:
Making a measurement on a state changes its wave function. No exceptions. None. The detached observer is not a reality in the quantum mechanical world.
3 Spin
I realized that the following sections will not make any sense if you don’t at least know what spin is. So, let’s make a short pit-stop at Spin City to learn about this nonsensical physical quantity.
We’re all aware of angular momentum– its the rotational analog of linear momentum (which we talked about the previous section). We all agree that it is a property related to the motion of an object, right? WRONG! Sometime in the 1900s (Seriously, 20th Century Physicists should chill out), it was discovered this angular momentum from motion i.e. “orbital” angular momentum, as it was called in the atomic physics context it was first described, does not account for all the angular momentum of a particle. Long story short, the remaining angular momentum, which is intrinsic to a particle, is now called Spin. Every fundamental particle has a particular value of spin, which, in quantum mechanical jargon, is the eigenvalue of the spin operator.
For understanding the following sections, we really only need to care about spin-1/2 particles, which are lovingly called fermions, and are the building blocks of all ordinary matter. The shining feature of spin-1/2 particles is that their spin can either be +1 or −1 , which is often referred to as spin-up (↑) and spin-down (↓) respectively.
Physically, the up or down comes from whether the measured spin is along the axis it is measured, or opposite to it. Yes, spin is a vector, so it does have three independent components in the three spatial directions, but it is convention to consider the z-component of the spin for calculations and experiments. Any references to up and down in the next sections are along the z-direction.
Oh, and one more thing, spin-0 particles have no intrinsic spin. This will be important when we encounter the EPR Paradox.
4 EPR Paradox
After skipping a whole bunch of most-likely important concepts in the study of quantum mechanics we arrive at the EPR paradox.
The EPR paradox is a thought experiment first described in the groundbreaking paper [1] by Einstein, Podolsky, and Rosen in 1935. Einstein was quite vocally a hater, and the EPR paradox was proposed as evidence that the description of reality provided by quantum mechanics is incomplete. Reality doesn’t care, of course, and the EPR Paradox isn’t really a paradox. In fact, it is the foundation of entanglement– a magnificent, very real feature of reality which spans black holes, quantum computers and even my field of research: Entanglement in elementary particle physics.
In fact, I’m so self-centered that the example we will use to illustrate the EPR paradox is from particle physics. Just kidding, my explanation follows Chapter 12 in Griffiths’ Introduction to Quantum Mechanics, and is a simplified version credited to David Bohm
EPRB Paradox
Suppose a pion (funky particle with spin-0) at rest, decays to an electron and positron which fly off into opposite directions. Since the pion has spin-0, conservation of angular momentum dictates that the electron and positron occupy the following spin configuration.
√(1/2) (|↑↓⟩−|↓↑⟩)
BE NOT AFRAID of the mathematical jumpscare. The fancy bracket |·⟩ is what’s called a “ket”, and is used to denote the state of a quantum system. All the expression says is that either the electron is
spin-up (+1) and the positron is spin-down (−1) or vice-versa, because the total spin of the system 22
must add up to 0. (Since the initial state is spin zero, the system must stay spin-zero even after the decay occurs. That’s what angular momentum conservation is all about.) We don’t know which combination we will get, but it must be one of the above. Measuring the spin of one of the particles will automatically tell us what the spin of the other particle is. This means that the spins of the electron and positron are correlated. In modern terms, such a state is called entangled.
Now, let’s pretend that these particles fly off in opposite directions, until say, they are several light years apart. What would happen if we found the electron and measured its spin to be +1 ? We instantly know that the positron’s spin is −1 . This is obvious. Why are we mad about this?
Naturally, we may think that the electron really was spin-up from the moment it was created and it was only that quantum mechanics did not know until we made a measurement. But by the principles of quantum mechanics, neither particle had a definite spin, until we made a measurement, causing the wave function to “collapse” and instanteously produce the spin of the positron which is lights years away!
The EPR bros were NOT having it. Einstein famously called this phenomenon “spooky action at a distance”. They stated that the quantum mechanical standpoint must be wrong! The electron and positron must have had well-defined spins from their creation, even if quantum mechanics does not know it. Quantum mechanics is not a complete description of reality and there must be some hidden variables which describe a physical system that we do not yet know.
The fundamental assumption guiding the EPR argument is that no information can propagate faster than light. This the principle of locality. In order to appease this, we can say that the wave function collapsed at some finite velocity and is not instantaneous. However, this violates conservation– If we measured the positron spin as well before the information of collapse reached it, there is a 50–50 chance that both particles are spin-up, which means the system has total spin-1. Preposterous! You can mess with anything you want in this universe, but you don’t mess with conservation laws. What do we do now?
Okay, let’s calm down. The theorists may say whatever they want, but experiment doesn’t lie. Experiment tells us that in these cases, spin is perfectly correlated. The wave function collapse is instantaneous. That’s crazy. Call your mom and tell her you want to go home. The EPR Bros are frightening you— Quantum Mechanics is NOT local so it is NOT complete.
...Except. It is. Enter, Bell’s Theorem.
5 Bell’s Theorem
Now, what’s the situation? The EPR gang is not happy. I’m not happy. You’re not happy. Is quantum mechanics wrong? No, silly! EPR said it themselves: they think it’s merely incomplete. So, in order to completely describe a quantum mechanical state, you not only need the wave function Ψ, you also need some unknown, hidden variable λ. Lots of hidden variable theories were proposed after the Einstein-Podolsky-Rosen paper, but none of them ever gained traction. It was still a respectable area of study until 1964, when J.S. Bell proved that any local (Remember locality from the last section?) hidden variable theory is incompatible with quantum mechanics.
I’ll spare you the details of Bell’s work, dear reader. One thought experiment in an essay is gruesome enough. (It is also getting quite late and I still didn’t code my calculations. I have spent far too much time on this already.)
Bell’s proof involves the wonderful use of probability, and the barest assumptions that can be made about local hidden variable theories. Basically, in any local hidden variable theory, the probabilities of various outcomes are related by what’s known as a Bell inequality. If EPR’s conjecture is right, and there really are hidden variables we don’t know about, then any physical system must obey its Bell inequality.
Except, there have been various experiments since the 1960s confirming that Bell’s inequality is indeed violated. This came as a rude shock to scientists as it is not fun to learn that reality is very much nonlocal. It was all fun and games when this was all merely a mathematical artifact, but nonlocality felt like a gateway drug to a much grimmer violation.
Causality
Bell inequality violations, no matter how surprising, are merely wonderful correlations between two sets of otherwise random data. Sure, the measurement of the spin of the electron affects the positron, but it does not cause it in any meaningful way. The person measuring the electron spin cannot use this collapse of the wave function to send a message to the person with the positron, since they don’t control the outcome of the experiment. They can decide whether to measure the electron at all, but the other person only has access to the positron’s spin and cannot tell whether the electron has been measured or not.
Phew! This sort of nonlocal influence does not transmit any energy or information, so it is exempt from the speed of light. Meanwhile, causal influences, those which do transmit information or energy, cannot travel faster than light. According to special relativity, if this was possible then, there are reference frames in which information can propagate backwards through time. And that, my dear reader, is what we call a big nono. Since the EPR paradox does not imply that causality is violated, we can lie uncomfortably on our bed of nonlocal but causal theory of quantum mechanics.
So rest easy, quantum mechanics is weird, but safe. Entanglement is not a fairytale, but also not the boogeyman. It’s probably more scared of you than you of it. Just give it some time. More answers will follow.
What Do I Do Now?
So, you want to know more? Or curl up in a ball and never think about this again? Either is fine. I won’t judge. If your answer is the former, here are some resources to guide you through the thickets of quantum mechanics.
PopSci Sources
1. IDTIMWYTIM: Heisenberg Uncertainty Principle 2. Why did Quantum Entanglement Win the Nobel Prize in Physics? 3. Bell’s Theorem: The Quantum Venn Diagram Paradox
Surely, you’ll get more out of these wonderful science Youtubers than you did from me yapping for four pages. There are a bunch more probably, but you’ll have to find them yourself.
Academic Sources
1. An Introduction to Quantum Mechanics, D.J. Griffiths.
Of course, there are other quantum mechanics textbooks that I like much more than this one. But, this is the least daunting, so I’ll leave it here.
Don’t forget to like and subscribe for more silly academic style papers.
References
[1] A. Einstein, B. Podolsky and N. Rosen, “Can quantum mechanical description of physical reality be considered complete”? Phys. Rev. 47, 777–780 (1935) doi:10.1103/PhysRev.47.777
[2] Heisenberg, W. “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”. Z. Physik 43, 172–-198 (1927). https://doi.org/10.1007/BF01397280
[3] D.J. Griffiths, D.F. Schroeter, “Introduction to Quantum Mechanics, Third Edition” Cambridge University Press (2018) 978–1–107–18963–8,
Dogs have had many jobs throughout history, in this case: Revenge.
“Those poor boys”
“She deserves to be punished too.”
“I’m not saying I support rape, but-”
“Sorry to say - she deserved it.”
“She put herself in harm’s way”
“But if she was fingered, then that’s not rape.”
“She ruined their lives.”
tfw = two fucking weiners
chillin on a Saturday night
let’s talk about todays geeked week teaser. we see vi being an absolute girl failure and jinx watching her sister beat up and get beaten up by people in the crowds of a grandstand and club. but caitlyn. caitlyn. vi clearly doesn’t see her again in her pit fighter arc. so i just got to thinking WHAT IF CAITLYN’s DEAD??? what if she went at the piltover-zaun war too hard and too wrecked with grief and lost her shit?? what if she pushed everyone away and went at it alone and got killed???? oh god i’m so scared
“i am a monument to all your sins” is such a fucking raw line for a villain it’s amazing that it came from halo, a modernish video game, and not some classical text or mythos
your gentle reminder that sevika can steal your bitch
“Those poor boys”
“She deserves to be punished too.”
“I’m not saying I support rape, but-”
“Sorry to say - she deserved it.”
“She put herself in harm’s way”
“But if she was fingered, then that’s not rape.”
“She ruined their lives.”
i can't vibe with anyone who thinks icarus was an ignorant idiot for flying too close to the sun. "oh i'd never do that i would have remembered my father's warning and been fine". do you seriously think that after years of imprisonment, feeling the sun on your face and the open air beneath your wings, you would be able to focus on anything but the joy of being alive and free? do you actually think that if you were given the opportunity to go where nobody has never been before, you wouldn't want to push it to the limit? to dare to be the first to try what no one else has ever even thought possible? do you honestly think you're too good for your own human nature? look me in the eyes and tell me if i strapped a pair of wings to your back that could take you wherever you wanted to go whenever you pleased that you'd be careful and sensible about it. you are not better than icarus just because you have the benefit of his example.
googledocs you are getting awfully uppity for something that can’t differentiate between “its” and “it’s” correctly
Oh shit I just realized I can post the "Gaussian Blur Wizard That Gaussian Blurs You" here
I wish lesbians were as easy to find in real life as they are on tumblr
Wait are we called mammals after mammary glands? Are mammals named after tits???
ARE WE THE BOOBS CLASS?
a cockroach sneaks into an art museum, ii
has anyone tasted the air before death strikes?
has anyone claimed onto canvas the moment before waves fall?
has anyone, like you, known intimately
the minute twist of emotions when
hot, fractious, unguarded wonder rots
into a disgust that flays you alive?
you have seen it run across the beetle-shine of your back
you have seen how the sight of your legs and your antennae, rigid with hair
latches onto happiness like a parasite
and flush it with terror and disgust
you have seen it all, the
nail-biting fear and fungus-green disgust
the technicolored nausea and the silver-sharp panic
you have seen the moment when waves fall
you have waited for the waves to fall
hold your breath and wait for the slightest hint of destruction-
but him, the young boy
the young artist with hands like violin strings, thrumming with potential-
and eyes full of hot, fractious, unguarded wonder
leaves you waiting
a wait that placidly trods far beyond its anticipated limit
and leaves you wondering
you have seen it all, the
fear and the disgust and the nausea and the panic
and you have tasted the air before death strikes
and the salt-spray before waves fall.
never have you tasted hope
never have you found a shard in time where you are the most human thing in somebody’s eyes
never have you become something worthy of wonder
something worthy of being known
a cockroach sneaks into an art museum by judas h. ( @judas-redeemed ) image id in alt
He isn’t used to this.
Most knights — if they have the gall (or status) to do so — approaches the princesses for their favour: beside him, Andromeda is already bestowing roses onto knights in shining, polished-until-reflective armour, Narcissa sniffs as she deigns to drop a flower onto the hands of one smarmy Lucius Malfoy, and Bellatrix glares at anyone who attempts to approach her, her kohl-lined eyes daring prospective suitors to just try her.
A normal sight in a joust. Or it would be, if not for the tall knight in dull silvery armour kneeling in front of Prince Sirius Black’s chair, asking for a rose of good luck.
His thoughts go faint. He’s simultaneously aware of everything and aware of nothing. Walburga and Orion’s faces burn with scandal. Regulus whispers an “oh my god”. The princesses, and the surrounding lords, beacons of colourful wealth amidst the silk-bannered good-natured cheer that festooned the arena, turn to stare. He senses doubt, incredulity, scandal, at the knight’s temerity, to seek a rose of good favour in unpolished, ragged armour, not revealing his face, not to a princess, but to a prince. A man. Heir to the throne.
Temerity indeed.
And yet the world mutes to a quiet, unremarkable buzz. The knight in front of him feels so very solid. So real. Through the gaps of his armour, chocolate eyes stare back at him, pools of rich brown (he picks out gold flecks in them like shimmery pepper) with intent written across every inch of the face he glimpses bits of through the face-mask. These eyes are staring at him, only.
“Give me your favour, my prince,” the knight repeats patiently, gazing intently into his eyes. Brown on pale blue. Oh yes. Temerity.
“Are you serious,” he mumbles faintly, letting the sentence trail off without a note of questioning.
Now the knight’s eyes sparkle with mirth; the intent wavers to allow laughter in. Sirius’ breath catches. “No,” he says. “You are.”
A joke on his name. Sirius allows the proud, haughty rule of his mouth to melt into a faint grin. Temerity indeed. Charming in a way it shouldn’t be. Nearby Regulus mutters something that sounds suspiciously like “Jesus Christ”.
Sirius decides this knight is fascinating. Slowly, staring straight into purposeful brown eyes, he untangles the tight seams holding his mouth into his usual hauteur, letting it loosen into a warm smile. He plucks a rose from Andromeda’s bouquet (he sees her smile) and slowly, very gently, ghosts his lips over the petals. A kiss of good luck. Prince Sirius’ favour. The corners of the knight’s eyes crinkle in endearment(?) and warm mirth. He presses the rose into the knight’s outstretched palms and truly allows his smile to grow wide.
“Good luck,” he murmurs.
“Thank you, my prince,” the knight answers in kind.
His mind is hazy. The world is quiet, like the knight has draped a veil over the noise and chatter. “Wait,” he says, feeling slightly foolish. (The thought shocks him. Sirius Black, feeling foolish? The world has indeed gone to shreds. The temerity of this knight to have caused this. Goodness.) “What’s your name?”
The knight’s eyes no longer crinkle in mirth — it’s more direct. Sultry. Sly, perhaps. “If I win,” he says, “you get to find out.”
Sirius stares at him, stunned.
The knight whisks back into the jostling crowd of competitors, framed in a whirl of colour and noise.
The quiet shatters and his world descends into chaos.
“That disgraceful knight—”
“His temerity—”
“Approaching a prince!”
“Is he a homosexual?”
“How dare he—”
“What were you thinking, honestly, giving him your favour—”
“And kissing the rose too?!”
“My goodness, Sirius—”
Regulus stares at him, green-grey eyes shining with something only siblings truly ever understand. A silent question. Are you sure you want to risk your heart for that guy?
He stares back. Thinking about that guy makes his heart buzz, and the world mute. His lips curve merrily, jauntily, but his eyes are just as intent as the knight’s. Yes, he answers silently. Yes.
~~~
As it happens, the knight wins.
Sirius lets out a breathless laugh as his family shakes their heads in disbelief. He knows how crazy it must look; an unnamed, unknown knight in hardly shining armour defeats the greatest warriors of the country? Novel. What temerity. He isn’t surprised, though. The purposeful gaze of the knight was one of utter intent and victory.
The knight raises a fist in victory, and tilts his head just so to meet Sirius’ eyes. He takes off his helmet.
Sirius’ breath catches for the hundredth time today.
Absolutely enchanting chestnut curls, tumbling gently to the neck and cheekbones, draping around those gold-flecked chocolate eyes. They look soft to touch, like velvet. Rich coffee skin, blanketing sharp cheekbones, a strong, square jaw, a straight and defiant nose. Scars that make him look devilishly charming streak across his nose bridge, his chin, his left cheek. Lips that look so pink, so full, wavering slightly with each puff of breath he takes.
Those lips smirk.
The knight — the very handsome knight that’s causing his heartbeats to hummingbird in Sirius’ chest — strides towards him, his good luck rose in hand. He stops directly in front of him, brings the red rose to those enrapturing lips, and delivers a kiss to the petals, directly above where Sirius kissed.
(He hears Walburga sniff in disdain and cannot find it within himself to care.)
The knight looks back down on him, eyes sparkling with not only mirth, with joy and with promise. “Thank you for your favour, my prince,” he says, and without the mask of armour his voice is deep, rich, edged with an absolutely sexy gravel, reminiscent of hot chocolate drunk on a cold rainy day. Sirius is entranced, and he feels his cheeks pinken. “My name is Remus Lupin.”
Fantasy AU where prince Sirius and his family attend a joust and a knight comes out in dull grey armour and people have no idea who he is but he goes straight to Sirius to hand him a rose for his luck in the tournament and when the knight wins he takes off his helmet to bow to the prince and it’s Remus